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Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
MA8151 [Regulation 2017]
Unit – I
Differential Calculus
1. Limit of a Function:
The limit of ( )f x , as x approaches a,
equals if we can make the value of
( )f x arbitrarily close to by taking
x to be sufficiently close to a but not
equal to a.
i.e. lim ( )x a
f x
2. Continuity:
A function f is continuous at a
number a if lim ( ) ( )x a
f x f a
.
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
3. Derivative:
The derivative of a function ( )f x at
x a denoted by ( )f a , is
0
( ) ( )( ) lim
h
f a h f af a
h
if this
limit exists.
(or) ( ) ( )
( ) limx a
f x f af a
x a
,
0
( ) ( )( ) lim
h
f x h f xf x
h
.
4. Table of derivative of the functions:
Sl.No. y dy
dx
1. Constant 0
2. nx 1nnx
3. x 1
4. 1
nx
1n
n
x
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
1
x
2
1
x
5. x
1
2 x
6. xe
xa
xe
logxa a
7.
log x
10log x
1
x
10
1log e
x
8. sin x cos x
9. cos x sin x
10. tan x 2sec x
11. cosecx cos cotecx x
12. sec x sec tanx x
13. cot x 2cosec x
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
14. 1sin x 2
1
1 x
15. 1cos x 2
1
1 x
16. 1tan x 2
1
1 x
5. Special Formulae:
(i) d dv du
uv u vdx dx dx
(ii) d dw du
uvw uv vwdx dx dx
dvuw
dx
(iii) 2
du dvv u
d u dx dx
dx v v
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
6. Equation of tangent line: 1 1( )y y m x x
7. Equation of normal line: 1 1
1( )y y x x
m
8. If the tangent line parallel to x-axis (horizontal) then 0
dy
dx .
9. If the tangent line parallel to y -axis (vertical) then 0
dx
dy .
10. Increasing and Decreasing Function
Let f be a function defined on the
interval [ , ]a b and have a finite derivative
inside the segment, then (i) f is increasing if and only if
( ) 0f x for all x in [ , ]a b .
(ii) f is decreasing if and only if
( ) 0f x for all x in [ , ]a b .
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
11. Monotonic Functions If a function f is completely increasing or
completely decreasing in an interval [ , ]a b
, then the function f is called monotonic
function in [ , ]a b . 12. Critical Number
A critical number of a function f is a
number c in the domain of f such that
( ) 0f c . 13. Maxima and Minima by First
Derivative Test Consider x a be a critical point of a
continuous function ( )f x .
(i) If ( )f x changes from positive to
negative at x a , then ( )f x has a
maximum at x a .
(ii) If ( )f x changes from negative to
positive at x a , then ( )f x has a
minimum at x a .
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
14. Maxima and Minima by Second
Derivative Test Consider x a be a critical point of a
continuous function ( )f x .
i) If ( ) 0f a , then ( )f x has a maximum
at x a .
ii) If ( ) 0f a , then ( )f x has a
minimum at x a . 15. Concavity Test
Suppose ( )f x is twice differentiable on an
interval I .
(i) If ( ) 0f x for all x in I , then the
graph of ( )f x is concave upward on I .
(ii) If ( ) 0f x for all x in I , then the
graph of ( )f x is concave downward on I .
16. Point of Inflection A point on a curve is called a point of P
inflection if the curve changes from
concave upward to concave downward or
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
from concave downward to concave
upward at . P
Unit – II
Functions of Several Variables
1. Euler’s Theorem:
If f is a homogeneous function of x
and y in degree n, then
(i) First Order
f fx y nf
x y
(ii) Second Order
2 2 2
2 2
2 22 1
f f fx xy y n n f
x x y y
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
2. Total Derivative:
If ( , , )u f x y z , 1 2( ), ( ),x g t y g t
3( )z g t then
du u dx u dy u dz
dt x dt y dt z dt
.
3. If 1 2( , ), ( , ), ( , )u f x y x g r y g r
then
(i) u u x u y
r x r y r
(ii) u u x u y
x y
4. Maxima and Minima:
Working Rules:
Step: 1 Find xf and yf . Put 0xf and
0yf . Find the value of x and y .
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
Step: 2 Calculate , ,xx xyr f s f yyt f .
Now 2rt s
Step: 3
i) If 0 , then the function have either
maximum or minimum.
1. If 0r f has maximum
2. If 0r f has minimum
ii) If 0, then the function is neither
Maximum nor Minimum, it is called
Saddle Point.
iii) If 0, then the test is inconclusive.
5. Maxima and Minima of a function using
Lagrange’s Multipliers:
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
Let ( , , )f x y z be given function and
( , , )g x y z be the subject to the
condition.
Form ( , , ) ( , , ) ( , , )F x y z f x y z g x y z ,
Putting 0x y zF F F F and then
find the value of , ,x y z .
6. Jacobian:
Jacobian of two dimensions:
, ( , )
, ( , )
u v u vJ
x y x y
u u
x y
v v
x y
7. The functions u and v are called
functionally dependent if ( , )
0( , )
u v
x y
.
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
8. ( , ) ( , )
1( , ) ( , )
u v x y
x y u v
9. Taylor’s Expansion:
1
( , ) ( , ) ( , ) ( , )1!
x yf x y f a b hf a b kf a b
2 21( , ) 2 ( , ) ( , )
2!xx xy yyh f a b hkf a b k f a b
3 2 2 31( , ) 3 ( , ) 3 ( , ) ( , ) ...
3!xxx xxy xyy yyyh f a b h kf a b hk f a b k f a b
where h x a and k y b
Unit – III
Integral Calculus
1.
1
1
nn x
x dx cn
2.
x xe dx e c
3.
1 1dx c
x x
1
1 1
1n ndx c
x n x
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
4.
3/22
3
xxdx c
5.
sin cosxdx x c
6.
cos sinxdx x c
7.
sec log sec tanxdx x x c
8.
cosec log cosec cotxdx x x c
9.
tan logsecxdx x c
10. cot logsinxdx x c
11. 1
2 2sin
dx xc
aa x
1
2sin
1
dxx c
x
12. 1
2 2cosh
dx xc
ax a
2 2( ) logor x x a c
13. 1
2 2sinh
dx xc
aa x
2 2( ) logor x x a c
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
14. 1
2 2
1tan
dx xc
a x a a
1
2tan
1
dxx c
x
15. 2 2
1log
2
dx x ac
x a a x a
16. 2 2
1log
2
dx a xc
a x a a x
17. 2
2 2 2 2 1sin2 2
x a xa x dx a x c
a
18. 2
2 2 2 2 1sinh2 2
x a xa x dx a x c
a
(or)
22 2 2 2log
2 2
x aa x x a x c
19. 2
2 2 2 2 1cosh2 2
x a xx a dx x a c
a
(or)
22 2 2 2log
2 2
x ax a x x a c
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
20. 2 2sin sin cos
axax e
e bxdx a bx b bxa b
21. 2 2cos cos sin
axax e
e bxdx a bx b bxa b
22. Reduction Formulae
2 2
0 0
cos (or) sin
1 3 5 2... .1 if is odd
2 4 3
n nxdx xdx
n n nn
n n n
1 3 5 1... . if is even
2 4 2 2
n n nn
n n n
2
0
sin cosm nx xdx
1 3 ... 1 3 ...
2 4 ...
m m n n
m n m n m n
1 3 ... 1 3 ...
2 4 ... 2
[ and are even]
m m n n
m n
m
m n n
n
m
23. 0
( ) 2 ( ) [if ( )isaneven function]
a a
a
f x dx f x dx f x
0 [if ( )isanoddfunction]f x
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
24. 0 0
( ) ( )
a a
f x dx f a x dx
25. ( ) ( )
b a
a b
f x dx f a x dx
26. Integration by Parts:
u dv uv vdu
27. Bernoulli’s Formulae:
1 2 3 4 ...uvdx uv u v u v u v
Unit – IV
Multiple Integrals
1.R
dxdy R
dydx Area (or)
To change into polar coordinate
cosx r siny r dxdy rdrd, and .
2.V
dxdydz V
dzdydx Volume (or)
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
Unit – V
Differential Equations
1. ODE with constant coefficients:
Solution C.F + P.Iy
Complementary functions:
Sl.No. Nature of
Roots C.F
1. 1 2m m ( )
mxAx B e
2. 1 2 3m m m 2 mx
Ax Bx c e
3. 1 2m m 1 2m x m x
Ae Be
4. 1 2 3m m m 31 2 m xm x m x
Ae Be Ce
5. 1 2 3, m m m 3( )
m xmxAx B e Ce
6. m i ( cos sin )x
e A x B x
7. m i cos sinA x B x
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
Particular Integral:
Type-I
If ( ) 0f x then . 0P I
Type-II
If ( )ax
f x e (or) sinhax (or) coshax
1.
( )
axP I e
D
Replace Dby a . If ( ) 0D , then it is P.I. If
( ) 0D , then diff. denominator w.r.t D and
multiply x in numerator. Again replace D
by a . If you get denominator again zero
then do the same procedure.
Type-III
Case: i If ( ) sin ( ) cosf x ax or ax
1. sin (or) cos
( )P I ax ax
D
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
Here you have to replace only for 2D not
for D . 2D is replaced by 2
a . If the
denominator is equal to zero, then
apply same procedure as in Type – I.
Case: ii If 2 2 3 3( ) (or) cos (or) sin (or) cosf x Sin x x x x
Use the following formulas 2 1 cos 2
2
xSin x
,
2 1 cos 2cos
2
xx
, x x x 3 3 1
sin sin sin 34 4
,
x x x 3 3 1cos cos cos 3
4 4 and separate 1 2
. & .P I P I
Case: iii If ( ) sin cos ( ) cos sinf x A B or A B ( ) cos cosor A B
( ) sin sinor A B
Use the following formulas:
1( ) in cos ( ) sin( )
2
1(ii) cos sin ( ) sin( )
2
1( ) cos cos cos( ) cos( )
2
1( ) sin sin cos( ) cos( )
2
i s A B sin A B A B
A B Sin A B A B
iii A B A B A B
iv A B A B A B
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
Type-IV
If ( ) mf x x
1.
( )
mP I xD
1
1 ( )
mxg D
1
1 ( ) mg D x
Here we can use Binomial formula as
follows:
i) 1 2 31 1 ...x x x x
ii) 1 2 31 1 ...x x x x
iii) 2 2 31 1 2 3 4 ...x x x x
iv) 2 2 31 1 2 3 4 ...x x x x
Type-V
If ( ) axf x e V where sin ,cos , mV ax ax x
1.
( )
axP I e VD
1
( )
axe VD a
Type-VI
If ( ) nf x x V where sin ,cosV ax ax
sin I.P of
cos R.P of
iax
iax
ax e
ax e
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
Type-VII
If ( ) sec (or) cosec (or) tanf x ax ax ax
1. ( ) ( )ax axP I f x e e f x dx
D a
1. ODE with variable co-efficient: (Euler’s Method) The equation is of the form
22
2( )
d y dyx x y f x
dx dx
Implies that 2 2( 1) ( )x D xD y f x
To convert the variable coefficients into
the constant coefficients
Put logz x implies zx e
2 2
3 3
( 1)
( 1)( 2)
xD D
x D D D
x D D D D
where dD
dx and
dD
dz
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
The above equation implies that
( 1) 1 ( )D D D y f x which is O.D.E with
constant coefficients.
2. Legendre’s Linear differential equation: The equation if of the form
22
2( ) ( ) ( )
d y dyax b ax b y f x
dx dx
Put log( )z ax b implies ( ) zax b e
2 2 2
3 3 3
( )
( ) ( 1)
( ) ( 1)( 2)
ax b D aD
ax b D a D D
ax b D a D D D
where dD
dx and
dD
dz
3. Method of Variation of Parameters: The equation is of the form
d y dya b cy f x
dx dx
2
2( )
1 2.C F Ay By and 1 2.P I Py Qy
where 2
1 2 1 2
( )y f xP dx
y y y y
and 1
1 2 1 2
( )y f xQ dx
y y y y
Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)
Textbook for Reference:
“ENGINEERING MATHEMATICS - I”
Publication: Sri Hariganesh Publications
Author: C. Ganesan
Mobile: 9841168917, 8939331876
To buy the book visit
www.hariganesh.com/textbook
----All the Best----